(a) Bisection Method:
This is one of the simplest and reliable iterative methods for the
solution of nonlinear equation. This method is also known as
binary chopping or half-interval method. Given a function
![$ f(x)$](img36.png)
which is real and continuous in an interval
![$ [a,b]$](img37.png)
and
![$ f(a)$](img38.png)
and
![$ f(b)$](img39.png)
are of opposite sign i.e.
![$ f(a)f(b)<0$](img40.png)
, then there is at
least one real root
![$ \xi \epsilon [a,b]$](img41.png)
of
![$ f(x)$](img36.png)
.
Algorithm:
Given a function
![$ f(x)$](img36.png)
continuous on a interval
![$ [a,b]$](img37.png)
satisfying the Bisection method starting criteria, carry out the
following steps to find a root
![$ \xi$](img42.png)
of
(1) Set
(2) For n=1,2,...until satisfied do
(a) If
![](img44a.gif)
(b)If
otherwise
Note:
1) The subscripts in
![$ a_{0},\quad b_{0},\quad a_{n-1},\quad
b_{n-1},\quad a_{n},\quad b_{n}$](img48.png)
etc denote the iteration number.
![$ [a_{0}, b_{0}]$](img49.png)
is the interval for the zeroth or starting
iteration.
![$ [a_{n}, b_{n}]$](img50.png)
is the interval for the n
![$ ^{th}$](img51.png)
iteration.
(2) An iterative process must be terminated at some stage. 'Until
satisfied' refers to the solution convergence criteria used for
stopping the execution process. We must have an objective criteria
for deciding when to stop the process. We may use one of the
following criteria depending on the behavior of the function
(monotous/steep variations/increasing /decreasing)
(i)
![$ \displaystyle {\vert x_{n+1}-x_{n}\vert<\epsilon_{a}}$](img52.png)
(Tolerable absolute error in
(ii)
![](img54a.gif)
(Tolerable relative
error in
(iii)
![$ \displaystyle {\vert f(x_{n+1}\vert<\epsilon } $](img55.png)
(Value of
function as
![$ x_{n}\rightarrow\xi$](img56.png)
)
(iv)
![$ \vert f(x_{n+1})-f(x_{n})\vert<\epsilon$](img57.png)
(difference in two
consecutive iteration function values)
Usually
![$ \epsilon_{a},\quad \epsilon_{r},\quad \epsilon$](img58.png)
are
referred to as tolerance values and it is fixed by us depending on
the level of accuracy we desire to have on the solution. For
example
![$ \epsilon_{a}\approx 10^{-70},\quad \epsilon_{r}\approx
10^{-12},\quad \epsilon\approx10^{-10}$](img59.png)
etc.
Example:
Solve
![$ 2x^{3}-2.5x-5=0$](img60.png)
for the root in the
interval [1,2] by Bisection section method.
Solution:
Given
![$ f(x)=2x^{3}-2.5x-5$](img61.png)
on
![% latex2html id marker 2526
$ \therefore$](img66.png)
There is a root for the given function in [1,2].
Set
![](img69a.gif)
Set
![$ \because\quad \displaystyle
{f(a_{1})f\left(\frac{a_{1}+b_{1}}{2}\right)=-2.6875< 0 }$](img72.png)
and
Set
Details of the remaining steps
are provided in the table below: